Outside the Box already enjoys a well-earned reputation as one of the best young bands in Asbury Park’s fecund music scene. But when the band was asked to do its third summer residency — a series of Wednesday night shows — at the Stone Pony, singer-guitarist Jeff Cafone wanted to give the band’s fans something special.

“We felt like if we were going to do this again, we had to kick it up a notch somehow,” he said, speaking before the band’s show on Wednesday. “So we decided to pay tribute to some of the artists that helped us become who we are today.”

This Wednesday, the talented quartet will perform Bob Dylan’s groundbreaking “Blonde On Blonde” in its entirety. For the first night of this series on June 1, the spotlight was on Elvis Costello’s debut album, “My Aim Is True,” with guest vocals from longtime supporter John “Southside Johnny” Lyon. And this past week, the band delighted an audience of mostly teenage fans (and a smattering of parents) with its rendition of Tom Petty and the Heartbreakers’ 1979 classic “Damn the Torpedoes,” sandwiched between short sets of originals.

Cafone and keyboardist Mark Masefield formed Outside the Box in 2004, when they were still in high school; the initial version of the band provided a youthful take on classic blues. Over the years, the rhythm section has changed a few times — bassist Ryan Wheeler and drummer Francis Valentino anchor the current lineup — and the band’s sound has matured to incorporate a Springsteen-like barroom swagger.

Cafone’s gritty vocals and supple guitar leads still reference the blues, while Masefield’s classic Hammond B3 organ and barrelhouse electric piano provide soulful undertones. But on originals such as “Love Is the New Villain” (just released as a single), “Safe Tonight” and the crowd sing-along “Forbidden Romance (Silenced the Radio),” Outside the Box crafts a sound that captures the heart and soul of Asbury Park’s musical traditions while still forging an identity of its own.

“Damn the Torpedoes” proved a brilliant choice to showcase OTB’s strengths, especially Masefield’s double-duty keyboards and Cafone’s effortless leads. The band nailed the familiar singles — the biting “Refugee” and the new-wavey “Don’t Do Me Like That.” But the highlights of the set came from some of the deep album tracks, which Cafone infused with his own personality.

This wasn’t Outside the Box doing Tom Petty karaoke; the boyish, curly-haired Cafone never tried to imitate Petty’s distinctive vocal style or mimic his guitar parts. Instead, he made the songs his own, while still capturing the essence of the originals. You could hear Petty’s debt to Lou Reed on the undulating “Here Comes My Girl” and the glammy Bowie-esque shadings of “Century City.” And with Masefield providing sonorous organ undertones, the band captured the country-western flavor at the heart of “Louisiana Rain.”

After a few more originals, Cafone invited the crowd back to hear Outside the Box’s take on “Blonde On Blonde”; then, as a tribute to Dylan, the band ended the night with a triumphant cover of “Like a Rolling Stone,” with Cafone transforming the anthemic chorus into a playful, jammy roundelay.

Outside the Box is currently completing work on its first full-length album, which should be released next month. Later in the summer, the band plans to perform the Band’s “The Last Waltz” and put together a night dedicated to the music of John Mayer.